Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 18/en

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Warm-up-exercises

Exercise

Prove the following properties of the hyperbolic sine and the hyperbolic cosine


Exercise

Show that the hyperbolic sine on is strictly increasing.


Exercise

Prove that the hyperbolic tangent satisfies the following estimate


Exercise

Prove by elementary geometric considerations the Sine theorem, i.e. the statement that in a triangle the equalities

hold, where are the side lengths of the edges and are respectively the opposite angles.


Exercise

Compute the determinants of plane and spatial rotations.


Exercise

Prove the addition theorems for sine and cosine using the rotation matrices.


Exercise

We look at a clock with minute and second hands, both moving continuously. Determine a formula which calculates the angular position of the second hand from the angular position of the minute hand (each starting from the 12-clock-position measured in the clockwise direction).


Exercise

Prove that the series

converges.


Exercise

Determine the coefficients up to in the series product of the sine series and the cosine series.


The next tasks require the definition of a periodic function.


A function

is called periodic with period , if for all the equality
holds.


Exercise

Let

be a periodic function and

any function.

a) Prove that the composite function is also periodic.

b) Prove that the composite function does not need to be periodic.


Exercise

Let

be a continuous periodic function. Prove that is bounded.




Hand-in-exercises

Exercise (3 points)

Prove that in the power series of the hyperbolic cosine the coefficients are if is odd.


Exercise (6 points)

Prove that the hyperbolic cosine is strictly decreasing on and strictly increasing on .


Exercise (4 points)

Let

be the space rotation by degree around the -axis counterclockwise. How does the matrix describing with respect to the basis

look like?


Exercise (5 points)

Prove the addition theorem

for the sine using the defining power series.


Exercise (4 points)

Let

be periodic functions with periods respectively

and . The quotient is a rational number. Prove that is also a periodic function.


Exercise (5 points)

Consider complex numbers lying in the disc with center and radius , that is in . Prove that there exists a point such that




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